Research on Dynamical Systems, Chaos and Fractals Modeled by Billiard Systems in Aspect of Global Analysis

全局分析方面台球系统建模的动力系统、混沌和分形研究

基本信息

批准号：
13440056
负责人：
KUBO Izumi
金额：
$ 7.23万
依托单位：
Hiroshima Institute of Technology (2003)Hiroshima University (2001-2002)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

We have improved the proof of the existence of Lipschitz continuous invariant foliations for two dimensional dispersing billiards without eclipse and make it more constructive and more elementary. This may give a new progress in applications. We introduced the geometry of geodesies due to Busemann to study the billiard ball trajectories in their configuration spaces. In particular, we found out some relations between parallels and periodic trajectories. Moreover, we observed interesting phenomena of billiard systems in a closed curve given by |x/a|^r+|y/b|^r=1(r>1) with computer simulations. Its Poincare map gives a lot of information, by which those phenomena may be proved.In hyperbolic and intermittent maps regarded as toy models of classical periodic billiards, transports are found to be characterized by the spectra of the Frobenius-Perron operator. Furthermore, we researched the relationship between boundary elements method and scattering problems in quantum billiards.We studied the renormalization associated to indifferent periodic points of complex dynamics. We were able to define a new space of maps which is invariant under parabolic renormalization and its perturbations.We observed large deviations for countable to one piecewise invertible Markov systems. In particular, we showed the(level 2)upper large deviation bounds and exponential decreasing property under certain conditions. Moreover, we researched multifractal version of large deviation laws.Further, we have succeeded to prove the conjecture by Boyle and Maass. We can construct a one-parameter family of complex structures with critical point at the point the recurrence and the transience switch to the other. We showed that freedom of deformation to preserve recurrence at the point is relatively low. As related topics to Quantum Chaos and so on, we have been studying the theory of operator algebras itself and obtained several results in the subject.

我们改善了Lipschitz的存在的证据，用于二维分散的台球，而无需蚀，使其更具建设性和更基本。这可能会在应用程序方面带来新的进展。由于Busemann，我们引入了大地测量的几何形状，以研究其配置空间中的台球轨迹。特别是，我们发现了相似之处与周期轨迹之间的一些关系。此外，通过计算机模拟，我们在| x/a |^r+| y/b |^r = 1（r> 1）的封闭曲线中观察到了台球系统的有趣现象。它的繁殖图提供了很多信息，可以证明这些现象。在双曲线和间歇性地图中，被视为经典周期性台球的玩具模型，发现运输是由Frobenius-Perron操作员的光谱所表征的。此外，我们研究了边界元素方法与量子台球中的散射问题之间的关系。我们研究了与复杂动力学无关的周期点相关的重新归一化。我们能够定义一个新的地图空间，该空间在抛物线重质化及其扰动下是不变的。我们观察到了可数到一个零散可逆的马尔可夫系统的巨大偏差。特别是，我们在某些条件下显示了（2级）上部较大偏差边界和指数降低的性质。此外，我们研究了大型偏差定律的多型版本。我们可以在复发点和瞬态切换到另一个方面构建具有临界点的复杂结构的单参数家族。我们表明，在该点保持复发的变形自由相对较低。作为与量子混乱相关的主题，我们一直在研究操作者代数本身的理论，并在该主题中获得了一些结果。